On a generalization of the Dirac bracket in the De Donder-Weyl Hamiltonian formalism

2015

Abstract.We discuss the precanonical quantization of fields which is based on the De Donder-Weyl (DW) Hamiltonian formulation and treats the space and time variables on an equal footing. Classical field equations in DW Hamiltonian form are derived as the equations for the expectation values of precanonical quantum operators. This field-theoretic generalization of the Ehrenfest theorem demonstrates the consistency of three aspects of precanonical field quantization: (i) the precanonical representation of operators in terms of the Clifford (Dirac) algebra valued partial differential operators, (ii) the Dirac-like precanonical generalization of the Schrödinger equation without the distinguished time dimension, and (iii) the definition of the scalar product for calculation of expectation values of operators using the precanonical wave functions.

“…The terms withĤ yield 20 Therefore, the first DW Hamiltonian equation in (64) is fulfilled on the average if the remaining three terms in (73)…”

Section: Ehrenfest Theorem In Curved Space-timementioning

confidence: 99%

Ehrenfest Theorem in Precanonical Quantization

2015

“…In this formulation, the Poisson brackets are defined on differential forms representing the dynamical variables. The construction of the brackets uses the polysymplectic structure (whose integration over the initial data yields the standard symplectic 2-form on an infinite-dimensional phase space of a field theory) related to the Poincaré-Cartan form in the calculus of variations and it leads to the Poisson-Gerstenhaber algebra structure generalizing the usual Poisson algebra known in the canonical Hamiltonian formalism [19,27,28] (for further generalizations see also [29][30][31][32][33][34]). The existence of a Hamilton-Jacobi theory corresponding to the DW Hamiltonian formulation [24][25][26] inevitably raises the question as to which formulation of the quantum theory of fields would reproduce the (partial derivative!)…”

Section: Introductionmentioning

confidence: 99%

Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization

Int. J. Geom. Methods Mod. Phys.

2019

The functional Schrödinger representation of a scalar field on an n-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical quantization based on the space-time symmetric De Donder-Weyl Hamiltonian theory. The functional Schrödinger representation emerges from the precanonical quantization when the ultraviolet parameter κ introduced by precanonical quantization is replaced by γ 0 δ inv (0), where γ 0 is the time-like tangent space Dirac matrix and δ inv (0) is an invariant spatial (n − 1)-dimensional Dirac's delta function whose regularized value at x = 0 is identified with the cutoff of the volume of the momentum space. In this limiting case, the Schrödinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration and the canonical functional derivative Schrödinger equation is derived from the manifestly covariant Dirac-like precanonical Schrödinger equation which is independent of a choice of a codimension-one foliation.

“…Let us recall that precanonical quantization [16][17][18][19] is based on the De Donder-Weyl (DW) Hamiltonian theory [23] which treats space-time variables on equal footing. In this formulation, the Poisson brackets are defined on differential forms representing the dynamical variables, that leads to the Poisson-Gerstenhaber algebra structure generalizing the Poisson algebra in the canonical Hamiltonian formalism [19,24] (see also [25][26][27]). The result of its quantization (a small Heisenberg-like subalgebra of it) is that both the operators and wave functions are Clifford-algebra-valued, and the precanonical Schrödinger equation includes the space-time Dirac operator instead of the standard time derivative.…”

Section: Introductionmentioning

confidence: 99%

SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

Reports on Mathematical Physics

2018

A relation between the precanonical quantization of pure Yang-Mills fields and the standard functional Schrödinger representation in the temporal gauge is discussed. It is shown that the latter can be obtained from the former when the ultraviolet parameter κ introduced in precanonical quantization goes to infinity. In this limiting case, the Schrödinger wave functional can be expressed as the trace of the Volterra product integral of Clifford-algebra-valued precanonical wave functions restricted to a field configuration, and the canonical functional derivative Schrödinger equation together with the quantum Gauß constraint can be derived from the Dirac-like precanonical Schrödinger equation.